Conformal duality of the nonlinear Schr\"odinger equation: Theory and applications to parameter estimation

The nonlinear Schr\"odinger equation (NLSE) is a rich and versatile model, which in one spatial dimension has stationary solutions similar to those of the linear Schr\"odinger equation as well as more exotic solutions such as solitary waves and quantum droplets. Here we present the unified theory of the NLSE, showing that all stationary solutions of the local one-dimensional cubic-quintic NLSE can be classified according to a single number called the cross-ratio. Any two solutions with the same cross-ratio can be converted into one another using a conformal transformation, and the same also holds true for traveling wave solutions. Further, we introduce an optimization afterburner that relies on this conformal symmetry to substantially improve NLSE parameter estimation from noisy empirical data. The new method therefore should have far reaching practical applications for nonlinear physical systems.